#6347163

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347145
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,21,23)(19,25,26,32)(20,28,30,24)(22,29,31,27);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19,21,26)(18,22,23,31)(20,29,30,27)(24,32,28,25);(20,30)(24,28)(25,32)(27,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,23)(19,26)(24,28)(27,29);(2,9)(7,15)(8,14)(10,16)(18,23)(22,31)(24,28)(25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,d^b=dh,f^b=fh,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254359);